We define the subspace homotopic distance between two continuous maps. This concept generalizes the idea of subspace Topological Complexity and Lusternik Schnirelmann category. Furthermore, we bound the homotopic distance of the ambient manifold with the sum of the subspace distances of the critical levels. This extends analogous results for the LS category and the TC. Moreover, we use this invariant to solve a generalized motion planning problem.

We study prime tensor ideals in tensor abelian categories of quiver representations. Specifically, we classify the prime tensor ideals in the category of representations of zigzag quivers (with bounded path length) whose vertex set is the set of integers. We show that prime tensor ideals in these categories are in canonical bijection with prime ideals of a Boolean algebra, the power set of integers.

A central topic of TDA is multi-parameter persistent homology. The existence of infinitely many non-intervals makes discrete complete invariants impossible. To address that challenge, we present a framework for extracting information in a fully commutative quiver via interval representations.

We propose an improvement to the graph representation learning method Node2vec via the application of persistent homology. This takes the form of additional loss functions given by the distance between the current embedding’s n-th dimensional persistence diagrams (PD) and some target PDs of the same dimensions. The minimization of these distances are implemented alongside the original Skip-Gram loss function of Node2vec. This improves the embeddings of graphs with multi-scale topological features, which we demonstrate with synthetic and real data examples.

Natural language processing (NLP) is a set of computational tools to process datasets consisting of a corpus of texts and to retrieve underlying information and features of the data. Researchers have incorporated artificial intelligence, machine learning, and data visualization into NLP techniques to facilitate analysis. Topological data analysis (TDA), specifically persistent homology, has been used to analyze these large datasets and extract topological features from the data. In this poster, we apply Latent Dirichlet Allocation (LDA) and TDA techniques to capture significant features such as patterns and themes in presidential election speeches during 2000-2020. We will also discuss potential future directions and implications for our work.

We propose a flexible approach for the detection of features in images with ultra low signal-to-noise ratio using cubical persistent homology. Our main application is in the detection of atomic columns and other features in transmission electron microscopy (TEM) images. Cubical persistent homology is used to identify local minima in subregions in the frames of nanoparticle videos, which are hypothesized to correspond to relevant atomic features. We compare the performance of our algorithm to other employed methods for the detection of columns and their intensity. Additionally, Monte Carlo goodness-of-fit testing using real-valued summaries of persistence diagrams---including the novel ALPS statistic---derived from smoothed images (generated from pixels residing in the vacuum region of an image) is employed to identify whether or not the proposed atomic features generated by our algorithm are due to noise.

We observe a signal composed of a certain number of reparametrized periods of a periodic function, corrupted by additive noise. We propose a topological method to segment the signal into its periods. We show that the persistence diagram of sublevel sets of a function is linear in the number of periods. We propose a method that estimates the number of periods correctly, if the extremal values of the function are generic enough.We construct an estimator by counting local extrema and we show that an estimator of the reparametrisation constructed from such a segmentation can be relevant in practice. We estimate the distance travelled by a vehicle from the measurements of the magnetic field measured inside that vehicle, by counting the revolutions of one of its wheels.

Classical Rips filtration is too sensitive to noise in point clouds, which is one of the problems in the practical use of topological data analysis. To address this issue, one often chooses a suitable weight on each point and uses the associated weighted Rips filtration. In this study, we propose a data-driven method to learn such weights based on the DeepSets architecture, which can be optimized through the differentiability of persistence diagrams. Numerical experiments show that our method performs better than classical methods in point cloud classification tasks in noisy settings.

The poster will first show the existence of the curse of dimensionality in persistence diagrams, which suggests the unreliability of using the observed persistence diagram in reality. Then, the poster will show the attempt to reduce/eliminate the curse of dimensionality by doing the normalized PCA.

In discrete Morse theory, the Morse complex of a manifold preserves homology while the complexity of computation is reduced. This thesis explains how Mischaikow and Nanda developed this approach for persistent homology. The second part explains how Bleher et al. (2022) applied persistent homology to identify critical mutations in the evolution of the coronavirus SARS-CoV-2. The discrete Morse algorithm for persistence computation is also implemented on this viral data set.

The magnitude is an invariant for metric spaces  introduced by Leinster. It measures the number of efficient points. The magnitude homology is defined by Hepworth and Willerton as a categorification of the magnitude. Recently, Asao and Izumihara introduced CW complexes whose homology groups are isomorphic to direct summands of graph magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres (of various dimensions) for cycle graphs with $2m-1$ $(m=3, 4, ¥cdots)$ vertices.

There is no denial that rapid and accurate personal identification and verification are crucial in the current society. Biometrics are data that can be used in these tasks. In particular, retinal fundus images have become a trustworthy biometric due to their uniqueness and stability over time. The standard way to analyse retinal images for such tasks are based on vascular and non-vascular features. This relies heavily on understanding the biological traits in these images.  Our approach, on the other hand, is purely mathematical and can be adopted to other biometrics as well. We proposed to use extended persistent homology (EPH) with respect to a radial filtration centred at optical disks to capture the topological features of the shape of retinal vessels.  To test the accuracy and speed of this approach, we applied the method to the retinal identification database (RIDB) which contains multiple samples of retinal images from 20 individuals. After summarising the EPH with persistence diagrams, we used a permutation test to examine whether it is possible to distinguish individuals based on the Wasserstein distances between corresponding persistence diagrams. The result is promising and we compare our approach with existing strategies by simulating scenarios and recording accuracy and processing times.

Discrete Morse Theory is, as the name may suggest, a combinatorial version of Morse Theory. In discrete Morse theory, topological properties of regular CW complexes $X$ are analysed by considering discrete Morse functions $f:X \rightarrow \mathbb{R}$. These topological properties can in turn be used to obtain cell decompositions of $X$ with fewer cells.

We give a brief introduction to the subject and focus on merge trees in discrete Morse theory. As in the smooth case, merge trees keep track of persistent $H_0$ information of the sublevel filtration induced by a discrete Morse function $f$. We define certain notions of equivalence between discrete Morse functions to investigate the inverse problem between discrete Morse functions on paths and trees and their induced merge trees. It turns out that discrete Morse functions on paths up to a notion of symmetry equivalence are in bijection to Morse labeled merge trees. Moreover, discrete Morse functions on trees up to component-merge equivalence are also in bijection to Morse labeled merge trees. Furthermore, a notion of order equivalence refines said bijections to Morse ordered merge trees.

At the end of the poster, we present future possible directions of the project.

 Rod packings are 3-periodic arrangements of lines in R^3 used by chemists to model structures formed by chain-like molecules such as polymers. We can associate a topological space to a rod packing by taking the complement, then quotienting by the translation symmetries to get a 3-torus with some holes in it. The fundamental group of this space gives a topological invariant of the rod packing, which has previously been calculated using the van-Kampen theorem in specific cases. We instead took a computational approach, using the discrete Morse theory program Diamorse to obtain cell structures for deformation retracts of these spaces. The resulting fundamental groups agree with those obtained by the original method. We also found 3-periodic nets corresponding to the 1-skeletons, one of which does not appear in the current databases.

Interdisciplinary interest in complex doubly periodic entangled structures, such as the weaves and polycatenanes considered here, motivates the development of new mathematical theories. In this poster, we will present some methodologies to build and encode their planar representations, using combinatorial and algebraic arguments on doubly periodic tilings of the plane. A generating cell of such a periodic entangled object can be seen as a particular type of link diagram embedded on a torus. This is a joint work with my supervisor Prof Motoko Kotani and Dr Mizuki Fukuda.

 For a reductive group G and a symmetric subgroup K, the direct product G/P × K/Q of partial flag varieties G/P and K/Q is called a double flag variety for a symmetric pair (G, K). Here, the diagonal action of K on G/P×K/Q is a vital object applied to branching rules of representations.

In particular, two problems are as follows:

(1)What are the pairs G, K, P, and Q such that there are only finitely many K-orbits on G/P×K/Q?

(2)Can we describe the K-orbits on G/P × K/Q when there are only finitely many K-orbits?

Therefore, we classified P and Q such that the number of K-orbits is finite for G=GL_{m+n} and K=GL_{m}×GL_{n}, and also described their orbit decomposition.

We solved the problem by providing a correspondence between the K-orbits and the quiver representations.

 Papillae on the surface of the tongue are integral to food oral processing, sensing and transport. Fungiform papillae contain taste buds responsible for taste perception, whereas filiform papillae are crucial for friction and textural perception. We propose a novel computational pipeline to detect papillae positions and classify fungiform and filiform papillae from a 3D mesh representation of a tongue surface. The approach  uses computational topology and intrinsic curvature profiles of papillae, combined with machine learning techniques. The first goal of this work is to remove the need for manual papillae identification and positioning, which are expensive and time-consuming. The second goal of this work is to quantify differences in papillae shapes between participants and in-person variability, i.e. how much do the shapes of fungiform and filiform papillae vary for one individual. The third goal is to determine if papillae can be used to predict the age, gender and participant id. The geometric, topological and curvature-based measurements extracted during the analysis could serve further for detecting a number of abnormal conditions which are linked to the tongue. This is pilot work with numerous further applications such as understanding papillae distribution and creation of artificial surfaces emulating physical properties of the tongue.

Glomeruli are bundles of capillaries through which blood is filtered in the kidneys, whose structure has been previously studied. One widely-described structural feature is lobularity—organization into strongly intra-connected lobes that are weakly inter-connected. Lobularity has been attributed to developmental processes and implicated in renal dysfunction but has not been rigorously defined. We propose a mathematical measure of lobularity and test whether it can distinguish biological glomeruli from a simplified model of capillary development. We traced mathematical graph models of 12 mouse glomeruli. Using circuit analysis to infer flow directionality and to represent each glomerulus as a Reeb graph, we then computed extended persistent homology.

Finally, we summarized the cycle features and their global-local outlier scores using skewness, the sharing statistic, and the Gini coefficient. We are using a random graph model based on mechanisms of angiogenesis to generate “null” distributions of these statistics. Our 12 glomeruli had 80–246 branchings and 118–367 vessels. Qualitative inspection of persistence diagrams revealed 1–3 exceptionally persistent cycles, indicating 2–4 lobes. Test statistics, for example Gini coefficients of cycle persistences (0.493±0.0529) and of outlier scores (0.707±0.0463), were narrowly distributed. Ongoing work compares these values to those computed for random angiogenic models to estimate p- values. Following this study, we will validate our conclusions on holdout data from manual encodings of previously diagrammed murine glomerular networks. Future work will seek a purely graph-theoretic measure of lobularity.